Constraint Learning by
Matching Distributions
Our approach to learning constraints in a generative
adversarial framework can also be interpreted as
matching the distribution over labels defined by the
model to the true marginal data distribution over the
labels. The former is specified implicitly by a sampling mechanism in which we first take a random
input x and obtain an output y by passing it through
a deterministic function f(x) (for example, a neural
network that outputs a sequence of object positions
given a set of input frames). Matching marginal distributions over the labels is a necessary condition for
the correct model, and it can be interpreted as a form
of regularization over the output space Y.

In slightly more formal terms, our constraint
learning framework can be seen as optimizing a
measure of similarity between distributions that can
be approximately computed through samples. Examples of such similarity measures include the Jensen-Shannon divergence or the Earth Mover’s distance.
Minimizing these divergences or distances is equivalent to training the model to satisfy the constraints
implicitly encoded in a set of representative output
samples.

Semisupervised Structured Prediction
Although constraint learning does not require a fully
labeled data set containing input-label pairs, providing
it with such data turns our problem into an instance of
semisupervised learning. In semisupervised learning,
we are given a small set of labeled examples and a large
unlabeled data set. The goal is to use the unlabeled data
to improve supervised accuracy.

Our constraint learning approach uses the smalllabeled data set to discover the high-level invariantsgoverning the system’s outputs and then uses theseinvariants to train the system on the large unlabeledset. In addition, we may combine our constraintlearning objective with a standard classification lossterm (over the labeled data), which acts as an addi-tional regularizer. This process can be interpreted asa new semisupervised algorithm for structured pre-diction problems, such as tracking, object detection,and pose estimation.

Traditional semisupervised learning methods
assume there is a large source of inputs x and tend to
impose regularization over the input space. Our
method, on the other hand, can exploit abundant
samples from the output space that are not matched
to particular inputs. Moreover, our method can be
easily combined with other approaches (Kingma et al.
2014; Li, Zhu, and Zhang 2016; Salimans et al. 2016;
Miyato et al. 2017) to further boost performance.

ExperimentsWe perform four experiments that demonstrate theeffectiveness of constraint learning in various real-world settings. We refer to the trained model as aregression network (or simply as a regressor) f, map-ping from inputs to outputs that we care about.

Our first two experiments use explicit constraints
in the form of formulas; the latter two rely on adversarial constraint learning, where we train an auxiliary
discriminator using output samples from a black-box
simulator. We refer the readers to our papers for network layout and training details (Stewart and Ermon
2017).

Tracking an Object in Free FallIn our first experiment, we record videos of an objectbeing thrown across the field of view and aim tolearn the object’s height in each frame. ExampleSPRING 2018 31